Dynamics of a Single-File Pore: Non-Fickian Behavior

The dynamics of an infinite one-dimensional system of hard rods is generalized to include the effects of a random background. Each rod follows a trajectory which is described by a generalized random function and when two rods collide they interchange velocities (and trajectories). An exact solution is obtained for the distribution $f_s(x,v,\frac{t}{v^{\prime }})$ which is the probability of finding a particle at $x$ with velocity $v$ at time $t$ that was at $x=0\mathrm{}$ with velocity $v^{\prime }$ at $t=0\mathrm{}$. The most interesting result is that in the long-time limit $p(x,t)\mathrm{}$, which is the probability of finding a particle at $x$ at time $t$ that was at $x=0\mathrm{}$ at $t=0\mathrm{}$, is of the form $t^{-\frac{1}{4}}\exp (-\frac{x^2}{t^{\frac{1}{2}}})$. Thus, the spatial distribution does not become Gaussian and Fick's law is not valid. It is suggested that this qualitative behavior might be expected whenever single-file effects become important and that it is not dependent on the details of the one-dimensional hard-rod collisions which have been used in the derivation.